# 2d transformation in computer graphics solved examples In this article, we will discuss about 2D Shearing in Computer Graphics. In a two dimensional plane, the object size can be changed along X direction as well as Y direction.

Given a triangle with points 1, 10, 0 and 1, 0. Apply shear parameter 2 on X axis and 2 on Y axis and find out the new coordinates of the object. Watch this Video Lecture. Get more notes and other study material of Computer Graphics. In this article, we will discuss about 2D Reflection in Computer Graphics. For homogeneous coordinates, the above reflection matrix may be represented as a 3 x 3 matrix as.

Given a triangle with coordinate points A 3, 4B 6, 4C 5, 6. Apply the reflection on the X axis and obtain the new coordinates of the object. Apply the reflection on the Y axis and obtain the new coordinates of the object. In this article, we will discuss about 2D Scaling in Computer Graphics. Given a square object with coordinate points A 0, 3B 3, 3C 3, 0D 0, 0. Apply the scaling parameter 2 towards X axis and 3 towards Y axis and obtain the new coordinates of the object.

In this article, we will discuss about 2D Rotation in Computer Graphics. Given a line segment with starting point as 0, 0 and ending point as 4, 4.

Apply 30 degree rotation anticlockwise direction on the line segment and find out the new coordinates of the line. We rotate a straight line by its end points with the same angle. Then, we re-draw a line between the new end points. Given a triangle with corner coordinates 0, 01, 0 and 1, 1. Rotate the triangle by 90 degree anticlockwise direction and find out the new coordinates. Transformation is a process of modifying and re-positioning the existing graphics.

In this article, we will discuss about 2D Translation in Computer Graphics. This translation is achieved by adding the translation coordinates to the old coordinates of the object as. Given a circle C with radius 10 and center coordinates 1, 4. Apply the translation with distance 5 towards X axis and 1 towards Y axis.

Obtain the new coordinates of C without changing its radius. Given a square with coordinate points A 0, 3B 3, 3C 3, 0D 0, 0. Apply the translation with distance 1 towards X axis and 1 towards Y axis. Obtain the new coordinates of the square. Computer Graphics.I still think we should use less trigonometry in computer graphics.

Various types of transformation are there such as translation, scaling up or down, rotation, shearing, etc. This transformation when takes place in 2D plane, is known as 2D Tutorials - teklastructures. Computerized Medical Imaging and Graphics. If you need to manage graphics, images such as JPEG, PNG, GIF images or pictures of any kind, or handle animation in your programs, including writing games, drawing 3D or 2D pictures, you might like to consider the graphics libraries, 3D engines, 2D engines, image manipulation source code etc listed here.

So essentially quaternions store a rotation axis and a rotation angle, in a way that makes combining rotations easy.

Reading quaternions. Computer Graphics Assignment Help, Rotation - 2-d and 3-d transformations, Rotation - 2-d and 3-d transformations Given a 2-D point P x,ythat we want to rotate, along with respect to an arbitrary point A h,k. What is a transformation? In this article, we will discuss about 2D Reflection in Computer Graphics. The reflected object is always formed on the other side of mirror.

## Translation

The size of reflected object is same as the size of original object. This way we can rotate by arbitrary angles, not just 90 degrees. Other Fun Matrices. Practice: Rotate 2D shapes in 3D. This is the currently selected item. Our mission is to provide a Tutorial 2D Rendering Being able to render 2D images to the screen is very useful. For example most user interfaces, sprite systems, and text engines are made up of 2D images. DirectX 11 allows you to render 2D images by mapping them to polygons and then rendering using an orthographic projection matrix.

With the drop in computer prices and the advent of the Java 2D API, high-end computer graphics are now available to all programmers, and for free! This is the second in a series of articles designed to introduce the Java 2D API to non-computer graphics experts. Current computer-aided design software packages range from 2D vector-based drafting systems to 3D solid and surface modelers. Modern CAD packages can also frequently allow rotations in three dimensions, allowing viewing of a designed object from any desired angle, even from the inside looking out.

Apr 13, Learning Outcome: 2D Transformation 1. Translation 2. Rotation 3. The Picture Dial VI combines a picture control with a rotating knob or dial control. Purpose of this Tutorial is to construct an intuitive bridge between our intuitions about 2D and 3D rotations and the quaternion representation.

The Tutorial will begin with an introduction to rotations in 2D, which will be found to have surprising richness, and will proceed to the construction of the relation between 3D rotations and quaternions.

Transformations in 2D, moving, rotating, scaling. The example with rotation around another point than the origin, can be realized like this in OpenGL: This material is described in most books on computer graphics.Rotate a triangle placed at A 0,0B 1,1 and C 5,2 by an angle 45 with respect to point P -1, The calculations available for computer graphics can be performed only at origin.

It is a case of composite transformation which means this can be performed when more than one transformation is performed. In this case since one of the edges of the triangle A is already at origin so after performing the transformation the values of A should not change, which will act as a check.

We will multiply the object matrix with the rotation matrix to get the solution. Remember: If direction of the rotation is not given then always assume it be positive. The more complex problem is handled in the example two where the rotation is happening with respect to a point other than origin.

S The triangle coordinates should also be written in the matrix form, shown as follows. The rotation matrix is also given below.

S Put the value of angle in the rotation matrix Angle given in problem statement is 45 degrees. Also put the values of cos 45 and sin 45 degrees in the matrix.

S Multiply these matrices to get the resultant matrix, which would describe the new coordinates of the edges of the triangle after 45 degrees rotation. Save my name, email, and website in this browser for the next time I comment. Wednesday, April 08, Example — 2 from Exams Rotate a triangle placed at A 0,0B 1,1 and C 5,2 by an angle 45 with respect to point P -1, The following composite transformation matrix would be performed as follows.

So tx and ty values would be negative. Perform the rotation of 45 degrees. Substitute of values of translation and rotation angle. The composite Transformation Multiply the resultant rotation matrix with the triangle matrix. The final resultant matrix will be as follows. Putting the values of rotation S Multiply these matrices to get the resultant matrix, which would describe the new coordinates of the edges of the triangle after 45 degrees rotation.

MCQs — Chapter Planning. MCQs — Chapter — Organizing. Related Posts. Leave a Reply Cancel reply Your email address will not be published. Comment Name Email Website Save my name, email, and website in this browser for the next time I comment.It is transformation which changes the shape of object. The sliding of layers of object occur. The shear can be in one direction or in two directions. Shearing in the X-direction: In this horizontal shearing sliding of layers occur.

The homogeneous matrix for shearing in the x-direction is shown below:. Shearing in the Y-direction: Here shearing is done by sliding along vertical or y-axis. Shearing in X-Y directions: Here layers will be slided in both x as well as y direction. The sliding will be in horizontal as well as vertical direction.

### 2D Transformation

The shape of the object will be distorted. The matrix of shear in both directions is given by:. JavaTpoint offers too many high quality services. Mail us on hr javatpoint. Please mail your requirement at hr javatpoint.

Duration: 1 week to 2 week. Computer Graphics. Projection Perspective Projection Parallel Projection. Next Topic Matrix Representation. Spring Boot. Selenium Py. Verbal A. Angular 7. Compiler D. Software E. Web Tech. Cyber Sec. Control S. Data Mining. Javatpoint Services JavaTpoint offers too many high quality services. The homogeneous matrix for shearing in the x-direction is shown below: Shearing in the Y-direction: Here shearing is done by sliding along vertical or y-axis.Transformation means changing some graphics into something else by applying rules.

We can have various types of transformations such as translation, scaling up or down, rotation, shearing, etc. When a transformation takes place on a 2D plane, it is called 2D transformation. Transformations play an important role in computer graphics to reposition the graphics on the screen and change their size or orientation. In this way, we can represent the point by 3 numbers instead of 2 numbers, which is called Homogenous Coordinate system.

In this system, we can represent all the transformation equations in matrix multiplication. A translation moves an object to a different position on the screen. The pair t xt y is called the translation vector or shift vector. The above equations can also be represented using the column vectors.

For positive rotation angle, we can use the above rotation matrix. To change the size of an object, scaling transformation is used. In the scaling process, you either expand or compress the dimensions of the object. Scaling can be achieved by multiplying the original coordinates of the object with the scaling factor to get the desired result. If we provide values less than 1 to the scaling factor S, then we can reduce the size of the object.

If we provide values greater than 1, then we can increase the size of the object. Reflection is the mirror image of original object. In reflection transformation, the size of the object does not change. The following figures show reflections with respect to X and Y axes, and about the origin respectively. A transformation that slants the shape of an object is called the shear transformation. There are two shear transformations X-Shear and Y-Shear. One shifts X coordinates values and other shifts Y coordinate values. However; in both the cases only one coordinate changes its coordinates and other preserves its values.

### 3D Transformation

Shearing is also termed as Skewing. The X-Shear preserves the Y coordinate and changes are made to X coordinates, which causes the vertical lines to tilt right or left as shown in below figure. The Y-Shear preserves the X coordinates and changes the Y coordinates which causes the horizontal lines to transform into lines which slopes up or down as shown in the following figure. If a transformation of the plane T1 is followed by a second plane transformation T2, then the result itself may be represented by a single transformation T which is the composition of T1 and T2 taken in that order.

Composite transformation can be achieved by concatenation of transformation matrices to obtain a combined transformation matrix.

## 2D Transformation

The change in the order of transformation would lead to different results, as in general matrix multiplication is not cumulative, that is [A]. The basic purpose of composing transformations is to gain efficiency by applying a single composed transformation to a point, rather than applying a series of transformation, one after another.A scaling transformation alters size of an object. In the scaling process, we either compress or expand the dimension of the object. The scaling factor s xs y scales the object in X and Y direction respectively.

Writing code in comment? Please use ide. Reverse a singly Linked List in groups of given size Set 3 Shortest Path Faster Algorithm Number of subarrays with GCD equal to 1 Find last remaining element after reducing the Array Maximum sum path in a Matrix Number of factors of very large number N modulo M where M is any prime number Minimize the maximum difference between adjacent elements in an array Minimum increment or decrement required to sort the array Top-down Approach Find lexicographically smallest string in at most one swaps Number of pairs such that path between pairs has the two vertices A and B Minimum Cost Graph Count of subarrays having exactly K distinct elements Traveling Salesman Problem using Genetic Algorithm Why Data Structures and Algorithms Are Important to Learn?

Minimum window size containing atleast P primes in every window of given range Maximum size of square such that all submatrices of that size have sum less than K Count the numbers with N digits and whose suffix is divisible by K Huffman Coding using Priority Queue Sum of GCD of all possible sequences Count maximum occurrence of subsequence in string such that indices in subsequence is in A.

Find triplet with minimum sum Find distance of nodes from root in a tree for multiple queries. Algorithm: 1. Make a 2x2 scaling matrix S as: S x 0 0 S y 2. For each point of the polygon. Draw the polygon using new coordinates. Below is C implementation:. Load Comments.Rotate a triangle placed at A 0,0B 1,1 and C 5,2 by an angle 45 with respect to point P -1, The calculations available for computer graphics can be performed only at origin.

It is a case of composite transformation which means this can be performed when more than one transformation is performed. In this case since one of the edges of the triangle A is already at origin so after performing the transformation the values of A should not change, which will act as a check. We will multiply the object matrix with the rotation matrix to get the solution.

Remember: If direction of the rotation is not given then always assume it be positive. The more complex problem is handled in the example two where the rotation is happening with respect to a point other than origin. S The triangle coordinates should also be written in the matrix form, shown as follows. The rotation matrix is also given below. S Put the value of angle in the rotation matrix Angle given in problem statement is 45 degrees. Also put the values of cos 45 and sin 45 degrees in the matrix.

S Multiply these matrices to get the resultant matrix, which would describe the new coordinates of the edges of the triangle after 45 degrees rotation. Save my name, email, and website in this browser for the next time I comment. Thursday, April 16, Example — 2 from Exams Rotate a triangle placed at A 0,0B 1,1 and C 5,2 by an angle 45 with respect to point P -1, The following composite transformation matrix would be performed as follows.

So tx and ty values would be negative. Perform the rotation of 45 degrees. Substitute of values of translation and rotation angle. The composite Transformation Multiply the resultant rotation matrix with the triangle matrix. The final resultant matrix will be as follows.

Putting the values of rotation S Multiply these matrices to get the resultant matrix, which would describe the new coordinates of the edges of the triangle after 45 degrees rotation. MCQs — Chapter Planning. MCQs — Chapter — Organizing. Related Posts. Leave a Reply Cancel reply Your email address will not be published. Comment Name Email Website Save my name, email, and website in this browser for the next time I comment.